We use lattice simulations to compute the baryon
spectrum of SU(4) lattice gauge theory coupled to dynamical
fermions in the fundamental and two-index antisymmetric (sextet)
representations simultaneously. This model is closely related to a
composite Higgs model in which the chimera baryon made up of
fermions from both representations plays the role of a composite
top-quark partner. The dependence of the baryon masses on each
underlying fermion mass is found to be generally consistent with a
quark-model description and large-Nc
scaling. We combine our numerical results with experimental bounds
on the scale of the new strong sector to derive a lower bound on
the mass of the top partner.

We have simulated the SU(4) lattice gauge theory coupled
to dynamical fermions in the fundamental and two-index
antisymmetric (sextet) representations simultaneously. Such
theories arise naturally in the context of composite Higgs models
that include a partially composite top quark. We describe the
low-lying meson spectrum of the theory and fit the pseudoscalar
masses and decay constants to chiral perturbation theory. We infer
as well the mass and decay constant of the Goldstone boson
corresponding to the non-anomalous U(1) symmetry of the model. Our
results are broadly consistent with large-Nc
scaling and vector-meson dominance.

We report preliminary results on the finite temperature
behavior of SU(4) gauge theory with dynamical quarks in both the
fundamental and two-index antisymmetric representations. This
system is a candidate to present scale separation behavior, where
fermions in different representations condense at different
temperature or coupling scales. Our simulations, however, reveal a
single finite-temperature phase transition at which both
representations deconfine and exhibit chiral restoration. It
appears to be strongly first order. We compare our results to
previous single-representation simulations. We also describe a
Pisarski-Wilczek stability analysis, which suggests that the
transition should be first order.

Models for what may lie behind the Standard Model often
require non-perturbative calculations in strongly coupled field
theory. This creates opportunities for lattice methods, to obtain
quantities of phenomenological interest as well as to address
fundamental dynamical questions. I survey recent work in this
area.